If one thinks of XX as a space of field configurations and of f=exp(iℏS)f = \exp(i \hbar S) as an exponentiated action functional, the one may think of this integral ∫exp(iℏS)μ\int \exp(i \hbar S) \mu as the finite-dimensional toy version of a path integral.

While this is in general not defined in the actual non-finite dimensional situations in field theory, the above re-formulation in terms of the chain homology of a BV-operator does make sense whenever an appropriate kind of differential is given. One may then try to axiomatize which chain complexes qualify as BV-complex and try to interpret their chain homology as computing perturbative path integrals.

on regular polynomial observables is, up to a factor of iℏi \hbar, the difference between the free component {−S′,−}\{-S',-\} of the gauge fixed global BV differential its time-ordered version (def. 1)

under the first brace that by assumption of a free field theory, {−S′,−}\{-S',-\} is linear in the fields, so that the first commutator with the Feynman propagator is independent of the fields, and hence all the higher commutators vanish;

by the product law for differentiation, where now ∇f≔(gab∂bf)\nabla f \coloneqq (g^{a b} \partial_b f) is the gradient and (v,w)≔gabvawb(v,w) \coloneqq g_{a b} v^a w b the inner product. Here one just needs to carefully record the relative signs that appear.

The same argument with the replacement 𝒯↔𝒯−1\mathcal{T} \leftrightarrow \mathcal{T}^{-1} throughout yields the other version of the equation (with time-ordering instead of reverse time ordering and the sign of the ℏ\hbar-term reversed).

Remark

(the “quantum shell”)

Beware that, superficially, it might seem that in equation (8) not only the term {−S′,𝒯−1(A)}\{-S',\mathcal{T}^{-1}(A)\} on the right vanishes on-shell, but also the term 𝒯−1{−S′,A}\mathcal{T}^{-1}\left\{ -S', A\right\} on the left, since the latter is the image under the linear map 𝒯−1\mathcal{T}^{-1} of an observable that vanishes on-shell.

To sort this out, notice that the isomorphism (9) tells us that the observables that vanish when passing from off-shell to on-shell observables are precisely those in the ideal generated by the image of {−S′,(−)}\{-S',(-)\}. But while 𝒯−1\mathcal{T}^{-1} is an isomorphism on (regular off-shell observables), it need not (and in general does not) preserve this ideal! Hence 𝒯−1({−S′,A})\mathcal{T}^{-1}(\{-S',A\}) need not (and in general is not) an element of that ideal, and this is why it remains when passing to the algebra of on-shell observables and thus makes its crucial appearance in the Schwinger-Dyson equation.